I have either heard or read that sharp asymptotics and bounds for oscillatory integrals of the form

$ \int e^{i \lambda \Phi(x)} \psi(x) dx \quad \lambda \to \infty $

are unknown when the critical points for the phase function are not isolated. If this impression is correct, what are the simplest / most important integrals of this form for which the optimal decay rate and asymptotic have not been proven? E.g. are there examples with $\Phi$ being a polynomial? (I would also appreciate recommendations of references for estimating multidimensional oscillatory integrals if anyone has them.)

2 Answers
2

As soon as the Hessian is not full rank, the problem becomes quickly messy:

if the hessian has rank $n-1$, then one can treat the one direction separately since we have explicit bound for a one-dimensional integral where the taylor expansion of $\Phi$ near a critical point $x_0$ looks like $(x-x_0)^p$ for any $p\ge 2$, the other directions will always give you $\lambda^{-\frac{1}{2}}$.

when the rank is less, then one must first identify those directions where the phase isn't quadratic, and look at the next terms in the expansion. V.I. Arnold then classifies the simple jets of functions in terms of their corresponding maximal decay in $\lambda$ in the following paper:

The classification is algebraic and does not rely on estimating integrals, so it never tells you how to obtain the estimate corresponding to that optimal decay. For the simpler classes of degenerate critical points, Popov has worked on estimating the oscillatory integrals:

Thanks for the references. I had actually never thought about oscillatory integrals from the point of view of numerical approximation. Right now what I'm mostly interested in is knowing which oscillatory integrals do not even have theoretical bounds that are sharp up to a constant. But I find these interesting.
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Phil IsettJul 27 '12 at 12:12